E1C03 09/14/2010
15:24:54 Page 109
COMMENT The magnitude of average value and amplitude of the 25 rad/s component of the
input signal are passed along to the output signal without much loss. However, the amplitude of the
400-rad/s component is severely reduced (down 61%). This is a filtering effect due to the
measurement system frequency response.
3.7 COUPLED SYSTEMS
As instruments in each stage of a measurement system are connected (transducer, signal conditioner,
output device, etc.), the output from one stage becomes the input to the next stage to which it is
connected, and so forth. The overall measurement system will have a coupled output response to the
original input signal that is a combination of each individual response to the input. However, the
system concepts of zero-, first-, and second-order systems studied previously can still be used for a
case-by-case study of the coupled measurement system.
This concept is easily illustrated by considering a first-order sensor that may be connected to a
second-order output device (for example, a temperature sensor–transducer connected to a recorder).
Suppose the input to the sensor is a simple periodic waveform, F 1 (t) ¼ A sin vt. The transducer
responds with an output signal of the form of Equation 3.8:
y t t
ð Þ ¼ Ce
Àt=t
þ
K t A
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ vt
ð Þ
2
q
sin vt þ F t
ð
Þ
F t ¼ Àtan
À1
vt
ð3:37Þ
where the subscript t refers to the transducer. However, the transducer output signal now becomes
the input signal, F 2 (t) ¼ y t , to the second-order recorder device. The output from the second-order
500
450
400
350
300
250
200
150
100
50
0
25
20
15
10
5
0
Amplitude
ω(rad/s)
(a) Input signal
500
450
400
350
300
250
200
150
100
50
0
25
20
15
10
5
0
Amplitude
ω(rad/s)
(b) Output signal
Figure 3.23 Amplitude spectrum of Example 3.12.
3.7 Coupled Systems 109
15:24:54 Page 109
COMMENT The magnitude of average value and amplitude of the 25 rad/s component of the
input signal are passed along to the output signal without much loss. However, the amplitude of the
400-rad/s component is severely reduced (down 61%). This is a filtering effect due to the
measurement system frequency response.
3.7 COUPLED SYSTEMS
As instruments in each stage of a measurement system are connected (transducer, signal conditioner,
output device, etc.), the output from one stage becomes the input to the next stage to which it is
connected, and so forth. The overall measurement system will have a coupled output response to the
original input signal that is a combination of each individual response to the input. However, the
system concepts of zero-, first-, and second-order systems studied previously can still be used for a
case-by-case study of the coupled measurement system.
This concept is easily illustrated by considering a first-order sensor that may be connected to a
second-order output device (for example, a temperature sensor–transducer connected to a recorder).
Suppose the input to the sensor is a simple periodic waveform, F 1 (t) ¼ A sin vt. The transducer
responds with an output signal of the form of Equation 3.8:
y t t
ð Þ ¼ Ce
Àt=t
þ
K t A
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ vt
ð Þ
2
q
sin vt þ F t
ð
Þ
F t ¼ Àtan
À1
vt
ð3:37Þ
where the subscript t refers to the transducer. However, the transducer output signal now becomes
the input signal, F 2 (t) ¼ y t , to the second-order recorder device. The output from the second-order
500
450
400
350
300
250
200
150
100
50
0
25
20
15
10
5
0
Amplitude
ω(rad/s)
(a) Input signal
500
450
400
350
300
250
200
150
100
50
0
25
20
15
10
5
0
Amplitude
ω(rad/s)
(b) Output signal
Figure 3.23 Amplitude spectrum of Example 3.12.
3.7 Coupled Systems 109